Monotonicity of Causal Killing Vectors and Geometry of ADM Mass Minimizers

We address two problems concerning the ADM mass-minimizing initial data sets. First, we show that the equality case of the positive mass theorem embeds into a pp-wave spacetime. Second, we show that positive Bartnik mass minimizers embed into strongly stationary vacuum spacetimes, thereby confirming the Bartnik stationary vacuum conjecture. A key ingredient is a new monotonicity formula for the Lorentzian length of a causal Killing vector field, which, among other applications, yields a strong maximum principle for the length.